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Non trivial identities

As you know there is a recurrent formula based on the cube trivial identities that holds true up to level 5 in QTM. It breaks at level 6:


P[0] = 1
P[1] = 12
P[2] = 114
P[3] = 1,068
P[4] = 10,011
P[5] = 93,840
P[6] < 879,624

The real number at level 6 is 878,880 which is 744 shorter than what the formula predicts.

This discrepancy is obviously caused by some identities other than the trivial ones.

Does anyone have the list of all those identities?

The only ones I know about are the ones listed in Jaap's website: